ON UNIQUENESS THEOREMS FOR ORDINARY
DIFFERENTIAL EQUATIONS AND FOR PARTIAL
DIFFERENTIAL EQUATIONS OF HYPERBOLIC
TYPE(X)
BY
J. B. DIAZ AND W. L. WALTER
1. Introduction. The present note contains proofs of uniqueness theorems
for the ordinary differential equation y' =f(x, y), and for the hyperbolic partial differential equation uxy=f(x, y, u, ux, uj), under what may be called
Nagumo uniqueness conditions. Reference is made to Kamke [5] for description and literature concerning the uniqueness condition on f(x, y) (which is
less restrictive
than the Lipschitz condition) which was introduced
into the
theory of the ordinary differential equation y'=f(x,
y) by Nagumo [l]. In
connection with the partial differential equation uxy=f(x, y, u, ux, uj), uniqueness conditions
more general than the Lipschitz condition were introduced
by Walter [7].
§2 deals with the ordinary differential equation, first under the assumption of a Lipschitz condition, and then a Nagumo condition. The argument
in the case of the Lipschitz condition is included because it appears to be
(under the assumptions
used) simpler and more direct than that currently
employed in textbooks,
which usually rely on the theory of the definite
integral to some extent (see, however, Caratheodory
[2], who treats a more
general case). The argument in the case of the Nagumo condition is also
independent
of the theory of the definite integral, and is to be compared with
the proofs in Kamke
[5] and Perron
[4]. §3 contains an extension of a
Nagumo type theorem of Walter
[7], for the characteristic
boundary
value
problem for uxy=f(x, y, u, ux,uj).
The main interest of the present considerations
seems to lie, not in the
results themselves (although one is led to the improvement
of known results,
e.g. the theorem in §3), but rather in the elementary
method employed,
which makes use only of the ordinary Lagrange mean value theorem of the
differential calculus and of simple properties of real valued continuous functions. The present approach appears to have been overlooked in the theory
of the subject. There is no doubt about the applicability of the method in
more general situations,
e.g. to systems of ordinary and partial differential
equations,
and to higher order equations,
which will not be considered ex-
plicitly here.
Received by the editors August 3, 1959.
(') This research was supported
by the United States Air Force through the Air Force
Office of Scientific Research of the Air Research and Development
Command under Contract
No. AF 49(638)-228.
90
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UNIQUENESS THEOREMS
91
2. The ordinary differential equation y'=/(x, y). (a) Lipschitz condition.
Suppose that the real valued function/(x,
y) is defined on the strip OSxSa,
— oo <y< + <x>,where a is positive. A "solution"
("in the classical sense") of
the ordinary differential equation y'=f(x, y) "on the interval O^x^o"
will
be understood
to be a real valued function y(x), continuous
in the closed
interval O^x^a
and possessing a finite derivative y'(x) throughout
the open
interval 0<x<a,
which satisfies y'(x)=/(x,
y(x)) ior 0<x<a
(i.e., a "solution" is required to be continuous in the closed interval and to satisfy the
ordinary differential equation in the open interval).
The function/(x,
y) will be said to satisfy a Lipschitz condition provided
that there is a number FSO such that |/(x, yf)—f(x, yf)\ ^F|yi —y2| when-
ever OSxSa
and — °° <yi, y2< + °°.
Theorem.
Let f(x, y) satisfy a Lipschitz condition on OSxSa,
— <x>
<y
< + <». Given a real number y0, there is at most one solution on OSxSa
of
fhe ordinary differential equation y'' =f(x, y) such that y(0) =yo-
Proof. Suppose that u(x) is a solution and that v(x) is also a solution.
Then one has u'(x)=f(x,
u(x)) and v'(x) =f(x, v(x)) on 0<x<a,
and w(0)
= v(0) =y0. Suppose, contrary to what one wants to prove, that the function
u(x)—v(x) (which is known to be continuous on OSxSo- and differentiable
on 0 < x <a and vanishes at x = 0) does not vanish throughout 0 S x S a. Then
there exists a number £ with OSi~<a, such that the absolute value function
\u(x)— v(x)\ =0 for 0^x^£,
and that \u(x) —v(x)\ is not identically zero
on any x interval i~SxS<~+lSa,
where />0. Choose />0 so small that /F<1,
where L is the Lipschitz constant of/(x, y). Since the function \u(x)— v(x)\
is continuous and not identically
zero on the closed interval %SxS£+l,
it
has a positive
maximum
at some number,
call it £m, of this interval
(notice
that £<£m). Now, applying first the mean value theorem of the differential
calculus to the function u —v on the interval £^x^£„,
and then using the
Lipschitz condition for/, one obtains, in succession, that
0 < | «(£„) - v(U | = | [re(U - v(U] ~ [«(£) - «(*)] |
= |U-£|
| «'(£*)-*>'(£*)|
= (£»-£) |/(£*,*(£*))-/(£*,*(£*))|
S IL | «(f) - v(?) | ,
where £* is the "mean value abscissa," satisfying
£<£*<£m^£-H;
and also
0 S IL < 1, the last by the choice of /. Thus the positive number | w(£m) —v(£m) |
is not the maximum value of the continuous function \u —v\ on the interval
a~SxS^+1,
which is a contradiction,
and the theorem is proved.
(b) Nagumo condition. A real valued function/(x,
y), of the type considered at the beginning of (a) above, will be said to satisfy a Nagumo condi-
tion provided
-
» <yi,
y2<
that
+
x|/(x,
yf)— f(x, y2)|^|yi
—y2| whenever
°°.
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OSxSa
and
92
J. B. DIAZ AND W. L. WALTER
Theorem.
[July
Let f(x, y) satisfy a Nagumo condition on OSxSa,
— <~<y
< + =o. Given a real number y0, suppose further that f is continuous in (x, y)
at (0, yj) (i.e., that lim(z,„),(o,„0)/(x, y) =/(0, yj)). Then there is at most one
solution on OSxSa
of the ordinary differential equation y' =f(x, y) such that
y(0) =yo.
Proof. As before, let u(x) be a solution and v(x) he a solution, on OSxSa.
Consider the function having the value \u(x)—v(x)\/x
on 0<xSa,
and the
value zero at x = 0. It will first be shown that this function is continuous on
the closed interval OSxSa
(the only real question is the continuity at x = 0).
(This function will be denoted just by \u(x)—v(x)\/x.)
Using the mean
value theorem of the differential calculus (notice that u —v vanishes at x = 0),
and the differential equations satisfied by u and v, it follows that, for 0 <x Sa:
| u(x) - v(x) I
I [u(x) - v(x)] -
X
[m(0) - v(0)] |
X
= I«'(**)
- v'(?)I
= I/(r,««•))-/cr,«(**))
I,
where 0<£*<x.
But/(x,
y) is continuous
\u(x) — v(x) I
UmAAI-AA
i-K)
X
at (0, yj), therefore
.
.
= | /(0) y0) - f(0, y0)| = 0,
as desired.
Now, suppose, contrary to what one wishes to prove, that the function
u(x)—v(x) is not identically zero on O^Sx^a. Then there exists a number
£ro>0 such that the function | u(x) —v(x) \ /x (which is continuous
on 0 SxSa
and is understood to have the value zero at x = 0) attains its positive maximum over 0^x^aatx
= £ro; and, furthermore
1u(x)- v(x)1
X
I „({,) - v(U)1
£m
whenever 0^x<£„.
However, by the mean value theorem of the differential
calculus, applied to the function u —v on the interval 0^x^£m,
and by the
Nagumo condition satisfied by/, one has that
;, I«(€_)- »(_.)I
1kf„) - KU] - [_(Q)
- v(Q)]
1
= | «'({*)" v'(?)I
= I/(?*,
-(«)-/cr, •«•))I
^ I«(r>- »(r)I
r
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1960]
UNIQUENESS THEOREMS
where the mean value
in direct contradiction
93
abscissa £* satisfies 0<£*<£„.
The last inequality
is
to the way in which £m was chosen, and the theorem
is proved.
Remark. The continuity requirement on/(x, y) made in the last theorem;
namely: lim(l,,,)..((>,„„)/(x, y) =/(0, yf), is essential for the validity of the
uniqueness
theorem.
This may be readily
seen from the following
Take yo = 0, for simplicity, and define/(x, y), for OSxSa,
1,
fix, y) = ' y/x,
. 0,
for
example.
by
y > x,
ior 0 < y S x,
for
y S 0.
The solutions are y = Cx, where 0 S CS 1. It is to be noticed that, in order for
the example to be valid, the term "solution" must be understood in the sense
specified at the beginning of this section. If this term, "solution," is relaxed
to mean that, in addition, y(x) has a finite right hand derivative at x = 0 (let
it be denoted by y'(0)) which satisfies the ordinary differential equation there
(i.e., y'(0) =/(0, y(0)); then, in this modified sense of the word, there is only
one solution through (0, 0) in the example. However, if the word "solution"
is understood
in this modified sense, then the theorem still holds, but the
hypothesis
concerning
the continuity
of / is now superfluous.
Because, if
u(x) is a solution, and v(x) is also a solution (in the modified sense of the
word) such that re(0) =v(0) =y<>, then it follows that limx,o |u(x)—v(x)| /x = 0
directly from the fact that w'(0) =v'(0), without using the continuity of/ at
(0, yo).
3. The partial differential
equation uxy=f(x, y, u, ux, uf). Suppose that the
real valued function/(x,
y, u, p, q) is defined for OSxSa,
0^y^6,
— oo
<u, p, </< + «>, where a and b are positive. A "solution" ("in the classical
sense") of the hyperbolic partial differential equation uxy=f(x, y, u, ux, uf)
"on the rectangle
O^x^ja,
OSySb"
will be understood
to be a real valued
function u(x, y) which is continuous, together with its two first order partial
derivatives
ux and uy, and its mixed second order partial derivative
uxy,
throughout
the whole closed rectangle O^x^a,
OSySb.
Clearly, by a classical theorem of H. A. Schwarz, any solution u(x, y) has the property that uyx
exists and equals u^ on O^x^o,
OSySb.
The function f(x, y, u, p, q) will be said to satisfy a Nagumo condition
provided that, for each (x, y) in the rectangle OSxSa,
OSySb,
there exist
numbers a(x, y) S 0, 8(x, y) S 0, y(x, y) S 0, with a(x, y) +8(x, y) +y(x, y) = 1,
such that
xy {fix, y, «i, pi, qf) - f ix, y, Ui, pi, qf) \ S a(x, y)\ux+ B(x,y)x\pi
whenever
u2\
- p2\ +y(x,y)y\qi
— » <uu u2, pi, p2, qu q2< + » ; and also
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-
q2\,
94
J. B. DIAZ AND W. L. WALTER
[July
x | f(x, 0, u, p, qj) - /(„, 0, u, p, qj) | g | oi - o21 ,
when
O^x^a,
— oo <„,
p, 51, a2< + oo, and
y I/(0, y, «, pi, a) - /(0, y, m,/>2,a) | S \ pi - Pi | ,
when O^y^o,
— =o <w, pi, p2, g< + <~. It is remarked that, if/(x, y, „, p, q)
happens to be, in addition, continuous
in the variable y at y = 0, for each
fixed (x, u, p, g), then the first inequality
in the definition of the Nagumo
condition
actually
implies the next inequality,
in which y = 0. A similar
statement applies to the inequality for x = 0 in the definition above. In particular, if/ is continuous for all (x, y, u, p, q) involved, and a, 0, y are constants, one has the Nagumo condition of [7].
Theorem.
Let f(x, y, u, p, q) satisfy a Nagumo
condition on OSxSa,
OSySb,
— co<&, p, g< + oo, with a(x, y), 0(x, y), y(x, y) continuous on the
closed rectangle OSxSa,
OSySb.
Given a real valued function a(x), defined on
OSxSa,
and a real valued function r(y), defined on OSySb,
with a(0) =t(0),
there is at most one solution on OSxSa,
Ogy^o,
of the partial differential
equation uxy=f(x, y, u, ux, uj) such that u(x, 0) =a(x),for
OSxSa
and u(0, y)
~T(y), for OSySb.
Remark.
tions
A similar
on the solutions,
uxy=f(x,
may
theorem,
be proved
but under much milder assump-
for the partial
differential
y, „), which is, in this sense, more an "analogue"
the equation
Proof.
general
uniqueness
uxy=f(x,
y, u, ux, uj) considered
(The argument
outline
of that
follows, with
of the theorem
equation
of y'=/(x,
y) than
here.
some essential
modifications,
in (b) of §2.) Suppose
that
the
u(x, y) is a
solution and that v(x, y) is also a solution on OSxSa,
OSySb. Then u(x, 0)
= v(x, 0)=a(x),
and ux(x, 0)=vx(x, 0)=a'(x),
for O^x^a.
It will be shown
next, using the result of (b) of §2, that also uy(x, 0)=i>„(x, 0) for O^x^a.
This follows at once from the remark that uy(x, 0) and vv(x, 0) are solutions
of the same ordinary
differential
equation
(in x) obtained
when one puts
y = 0 in the partial differential equations
satisfied by u and v; and also satisfy
the same "initial"
condition
at x = 0, namely uy(0, 0)=flv(0,
0)=t'(0).
Further, the "right hand side" of this ordinary differential equation:
w'(x) =/(x,
satisfies the Nagumo
condition
0, <r(x), <j'(x), w(x)),
of §2, because
x | /(x, 0, a(x), a'(x), w) — f(x, 0, <r(x), <r'(x), wj)\
tor OSxSa,
and
S \ w — Wi\
— oo <w, u»i< + oo. Hence uy(x, 0) =vy(x, 0), lor OSxSa,
desired.
Similarly, one has that
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as
1960]
UNIQUENESS THEOREMS
95
«(0, y) = v(0, y) = r(y),
Uy(0, y) = vy(0, y) = r'(y),
and that
ux(0, y) = vx(0, y),
ior OSySb.
u(x, y):
Consequently,
from the partial
differential
equation
satisfied by
Uxy(x, y) = f(x, y, u(x, y)ux(x, y), uy(x, y)),
(and from the corresponding
equation
for v(x, y)) it follows also that
Uxy(x, 0) = Vxy(x, 0),
for OSxSa,
and that
«*»(0, y) = *w(0, y),
for OSySb.
Next, consider the auxiliary function
writing) defined on the rectangle OSxSa,
a(x, y)-
(call it F(x, y) to abbreviate
the
OSySb,
which has the value
| u(x, y) - v(x, y) \
xy
, a,
,|
ux(x, y) -
+ B(x, y)-1-
vx(x, y)\
y(x, y)-
\ uy(x, y) -
y
vy(x, y) \
x
when 0<xSa,
and 0<ySb;
and has the value zero when either x = 0 and
OSySb,
or when O^x^o
and y = 0. It will now be shown that the auxiliary
function F(x, y) is continuous in the whole closed rectangle. Clearly, F(x, y)
is continuous on 0<xSa,
0<y^Z>, it only remains to verify that
lim
F(x, y) = 0,
0 S x S a,
lim
F(x, y) = 0,
0 S y S b.
and that
(*,»)—tfl.v)
Since the functions a, 8, and y are continuous
suffices to prove that the three functions
m(x, y) - v(x, y)
-,xy
where 0<x^a,
OSxSa,
0<y^&,
ux(x, y) -
y
approach
vz(x, y)
on the closed rectangle,
ancj
uv(x, y) - w(x, y)
-,
x
zero as (x, y) approaches
or as (x, y) approaches (0, y), with OSySb.
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it
(x, 0), with
96
J. B. DIAZ AND W. L. WALTER
[July
One has, by the ordinary mean value theorem of the differential calculus,
applied to the function u(x, y) —v(x, y), regarded as a function of y, for fixed
x>0
(recall that u(x, 0)-v(x,
0)=0),
that
u(x, y) - v(x, y)
uy(x, rj) - vv(x, -n)
xy
x
where 0<r;<y.
Now, applying the mean value theorem of the differential
calculus to the function uy(x, rj)—vy(x, rj), regarded as a function of x, for
fixed r] (recall that uy(0, ri)—vy(Q, 17)=0), one obtains
u(x,y)
-
-v(x,y)
xy
= «._(£,v) - »»*(?,v),
where 0<£<x,
0<77<y. (This last equation is a mean value theorem for the
differential operator d2/dydx, see Goursat [3].) Proceeding in a similar manner, one is led to the equality
Mx(x, y) — vx(x, y)
-
= u^x,
v*) - v^x, rj*),
y
where 0<7j*<y,
and to the equation
«!/(£> y) — %(x, y)
-=
x
uvx(t,
y) -
vyx(?,
y),
where 0<£*<x.
But the function uxy —Vxyis continuous on the closed rectangle and vanishes when x = 0 and when y = 0. Thus the three functions
(u —v)/xy, (ux —vj)/y and (uy —vj)/x tend to zero as asserted; and this, in
turn, implies the desired continuity
of the auxiliary function F(x, y).
Now for the remainder of the proof. Suppose, contrary to what one wants
to prove, that the function u(x, y)—v(x, y) is not identically
zero on the
closed rectangle OSxSa,
OSySb.
Then the auxiliary continuous function
F(x, y) described above must have a positive maximum at a point (£„,, 7]m)
such that 0<i-mSa,0<r)mSb;
and that, further, F(x, y) <CF(£m, r)m) whenever
OSx+y
<£m+77m with OSxSa
and OSySb.
(To see this, consider the set of
points (x, y) with OSxSa,
OSySb,
at which the value of F equals the positive maximum of F, and choose (£m, r)m) as a point of this set for which the
continuous
function x+y has a minimum, which must necessarily be positive.)
At this stage one gets a contradiction
by applying the mean value theorem twice to the a "term" of F($jm, t)m), once to each of the other two terms
(0 and 7) of F(£m, iym), and then using the Nagumo condition which is satisfied by/.
(It is to be recalled that the three functions u —v, ux —vx, and
uv —vv vanish for OSxSa,
y = 0 and for x = 0, Ogy^o.)
One obtains, for
example, that
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1960]
UNIQUENESS THEOREMS
| »(£>», Vm) — »(£m, Vm) \
,
-=
,.
97
,|
I «*»(£»17)- »*»(£>
f) I
=
|/(£,i7,
«(£,»),
«x(£, 17), «i,(£,i?))
- /(£, 17,»(£, 17), »x(£, 17), »»(£, 17))I
^
^
/t x I "(k^) ~ "(£,17)I
a(£,
1?)-
£17
,
+
. I «x(£, 17)- »z(£, 17)I
/?(£, 17) -
17
,
j re„(£, 7?) -
;>,,(£,n) |
+ y(£, 17)-=
where 0<£<£m
I «x(£m,l?m)
and 0<w<ijm.
— **(£m,17m) | =
=
F(£, v),
Similarly
| «x»(£m, 17*) ~
I /(£»,
»xV(£m, 17*) |
17*, «(£*, 17*), wx(£m, 1?*), «„(£„, 17*))
- /(£m, 17*, »(£», 17*), »»(£-», 17*), l>„(£m, 17*)) |
- „ « l«(6-^*)-"(Un*)l
= «(£-**)-rr*- KmV
, „,*. *-,I u*itm,v*)
- »«(£»,!;*)
I
+ j8(£m, 17*)-
1?*
..
„ \Uy(tm,V*)
+ y(£m, 17*)-=
~VV(UV*)\
^
F(£m, v*),
£
where 0<rj*<wm;
and also
I «v(£m, 17m) -
»„(£«, l?m) | S F(£*, 17m),
where 0<£*<£m.
Putting together the last three inequalities,
auxiliary function F, one obtains
and using the definition
of the
F(£m, i?m) S «(£„, Vm)F(S, v) + /8(fc», i7»)F(£m, 1?*) + y(£m, i?m)F(£*, Vm),
where £+17, £m+i7* and £* + i7m are less than £OT+77TO.
This contradicts
choice of (£m, r;m), since by the definition of (£m, rjm) one must have
F(£,i7)
F(U,f)
'
<F(U,Vm);
Fit*,Vm)\
while, at the same time
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the
98
J. B. DIAZ AND W. L. WALTER
«(&n,
[July
Vm)
0(U Vm) = 0,
y(U, Vm).
and
«(£m, Vm) +
/3(£m, Vm) +
y(£m, Vm) =
1,
and the proof is complete.
4. Concluding remarks.
It was mentioned in the introduction
that the
method of proof employed in this note is also applicable to more complicated
problems for ordinary differential equations and for partial differential equations of hyperbolic type. In lieu of the formulation of the general case, which
involves a nonlinear integral equation of Volterra type in several variables,
and is both cumbersome
and not transparent,
only several particular exam-
ples will be discussed.
(a) Consider the initial value problem for the ordinary differential equation of «th order: y(n)=/(x,
y, y', • ■ • , y<-*-v)-t OSxSa,
where the values
y(,)(0)=yv
(j>= 0, • • • , n — 1) are assigned. One can prove uniqueness with
a method which is essentially that of §2, provided that/ satisfies the following condition of Nagumo type:
n-l
x" | f(x, z, Zi, • • - , Zn-j) —f(x, 2, Ii, • • • , Zn-j) I S JZ (n — v) \a,(x)xy \ z, — z,\
i—O
for OSxSa
and
— oo <z, z, Z\, z\, ■ - ■ , zn-i, z„_i<
+ oo, where
the functions
„,(x) are supposed to be nonnegative
and continuous
on OSxSa
and
JZ"-o a,(x) = 1. (For the proof, use the auxiliary function which is zero for
x = 0 and equals JZ"-l (n~p)lct,(x)x'~n\
uM(x)— vM(x)\ for 0<xSja,
where
u(x) is a solution and v(x) is also a solution.)
(b) In the characteristic
initial value problem
for the equation
uxy
=f(x, y, z, u, ux, uy, uz, uxy, uxz, uyj), one looks for a solution u = u(x, y, z) in a
parallelepiped
O^x^a,
OSySb,
OSzSc,
which takes on prescribed
initial
data u(x, y, 0)=a(x,
y), w(0, y, z)=r(y,
z), and u(x, 0, z)=<p(x, z). In this
case the uniqueness
condition
of Nagumo type reads as follows:
xyz | f(x,
y, z, u, ux, uy, ut, uxt, uxx, Uyj) — f(x,
y, z, ii, ux, uy, uz, u^,
uxz, uyj) |
S _i | „ — m | + a2x I ux — ux | + azy | uy — uy \ + atf \ u, — uz\
+ caxy | u^ — u^ I + _exz | uxz — uxz \ + ct-, yz \ uyz — uyt \ ,
where a,- (*= 1, • • • , 7) are supposed to be continuous nonnegative functions
of (x, y, z) on O^x^a,
0^y^6,
O^z^c,
whose sum is equal to unity. If/
is required to be continuous in (x, y, z) for each (fixed) set of its remaining
independent
variables, then uniqueness can be proved along the lines of §3.
If this assumption about/is
not made, then in the formulation of the Nagumo
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1960]
UNIQUENESS THEOREMS
99
condition one needs additional inequalities, similar to those of §3, which ensure the uniqueness of the partial derivatives
ux, uy, uz, u^, uzt, uyz on the
three edges through (0, 0, 0) and on the faces x = 0, y = 0 and 2 = 0 of the
parallelepiped;
the exact formulation is left to the reader.
(c) An example of a higher order equation in several variables is uxxxy
=f(x, y, u, ux, uy, uxx, uxy, uxxx, uxxy). Here one looks (for example) for a solu-
tion u(x, y) in the rectangle OSxSa,
OSySb,
which assumes the following
prescribed
initial values: u(x, 0)=a(x)
and w(0, y)=r(y),
ux(0, y)=Ti(y),
«xx(0, y) =Ti(y), where <r(0) =t(0), a'(0) =n(0),a"(0)
=t2(0). Uniqueness
can
be proved if/satisfies
the following condition of Nagumo type:
x y I j\X,
y, u, ux, uy, uxx, u^,
S 6c*i | u — u | +
+
2a6xy
uxxx, uxxy)
2a2x \ux
| Mn, — Uxy | +
— ux\
f\x,
+
y, u, ux, uy, uxx, Uxy, uxxx, uxxy) j
6a3y \ uv — w„ | + a4x21 uxx — uxx \
atjX31 uxxx — Uxxx I +
«7X2y | uxxy — uxxy \ ,
where again the a, are continuous nonnegative functions and cti+ • ■ ■ +0:7
= 1. As in §2, this condition has to be accompanied by additional conditions
on the characteristics
x = 0 and y = 0 if / is not required to be continuous in x
and y when its remaining independent variables are held fixed.
(d) It should be noticed that systems of differential equations can be
treated in the same manner. If, for example, a system of ordinary differential
equations y' =/(x, y) is given, where y is a vector in F„: y = (yi, • • • , yf) and
/ is a vector-valued
function; and if a norm
||y||
is defined
= ci\ yx\ +
in Rn, then
• ■ • + cn\ yn\,
the uniqueness
(ct > 0, i = 1, • • • , re),
condition
of Nagumo
reads
exactly
as in §2: x||/(x, y)—f(x, y)\\ s\\y —y\\- The reasoning is precisely the same as
in §2, except that the single "mean value abscissa" £* may now be obtained
in the following slightly modified manner:
||m(U - v(U)\\
-=
1 A
,
,
1 ^
— 2w d I «.'(£"•)— Mfm) I = — 2L,e.c,[re,(£m)- »,(£m)J
X
X i_i
X ,_i
= i: Ciei[ui(n - *.•(£*)]S ||«'(£*) - »'(£*)||,
1-1
where each number
e, is either
+1 or —1, as the particular
circumstances
may demand.
(e) It was shown by Perron [4] that the uniqueness theorem of Nagumo
for the ordinary differential equation y' =/(x, y) is a "best" theorem in the
following sense. For every C>1 there exist bounded continuous functions
f(x, y) for which the inequality x|/(x, y)-f(x,
y)\ SC\y —y\ holds and for
which the initial value problem y'=/(x,
y); y(0)=y0,
has more than one
solution. In [7] one of the authors showed that the theorem of §3 is also a
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100
J. B. DIAZ AND W. L. WALTER
best theorem in a similar sense. An analogous statement can be made about
the more general uniqueness theorems indicated in this section. The construction of counter examples resembles that of [4] and [7].
Added in proof, May 10, 1960: At the conclusion of a lecture given by one
of us at RIAS, in Baltimore,
in October 1959, Professor Philip Hartmen
kindly drew our attention to results, related to those of §3 above, obtained
by his student,
Mr. J. Shanahan,
which had been submitted
to Mathe-
matische Zeitschrift.
Bibliography
1. M. Nagumo,
Eine hinreichende
Bedingung fur die Unitat der Losung von Differential-
gleichungen erster Ordnung, Jap. J. Math. vol. 3 (1926) pp. 107-112.
2. C. Caratheodory, Vorlesungen uber reelle Funktionen, 2d ed., Berlin, 1927, pp. 665-674.
3. E. Goursat, Cours d'analyse mathematique, 5 ed., vol. 1, 1927, pp. 42-43.
4. O. Perron,
Eine hinreichende
Bedingung fur die Unitat der Losung von Differentialgleich-
ungen erster Ordnung, Math. Z. vol. 28 (1928) pp. 216-219.
5. E. Kamke, Differentialgleichungen reeller Funktionen, Leipzig, 1930, pp. 96-100.
6. G. Sansone, Equazioni differenziali nel campo reale. II, Chapter 8, 2 ed., Bologna, 1949.
7. W. Walter, Eindeutigkeitssatze fur die Differentialgleichung
uxy =/(*, y, u, ux, uj), Technical note BN-159, Institute for Fluid Dynamics and Applied Mathematics,
University fo
Maryland, 1959. (See also Math. Z. vol. 71 (1959) pp. 308-324.)
University of Maryland,
College Park, Maryland
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